The Supreme Court sent a major affirmative action case back to a lower court for reconsideration on Monday, leaving many local college officials relieved that race-conscious admissions policies were not overturned, but anxious about how to meet the new, higher standard the justices set for their use.

In reviewing a challenge to affirmative action in admissions at the University of Texas, the justices did not lay out exactly how to meet that tougher standard, ensuring that university presidents, attorneys, and admissions officials will be puzzling over whether they are truly in compliance with the court’s standard.

Laurie Pohl, vice president for enrollment and student affairs at Boston University, said she was “somewhat relieved they didn’t do anything terribly damaging, but frustrated that we still know that there is a lot of work to do to continue to comply with” the law.

The case stems from a lawsuit that Abigail Fisher, who is white, filed against the University of Texas at Austin after her application to join the freshman class was rejected in 2008. She argued that the university’s consideration of race violated the equal protection clause of the 14th Amendment.

Writing for the 7-1 majority, Justice Anthony M. Kennedy found that the appeals court that rejected Fisher’s argument failed to meet the legal requirement that it apply “strict scrutiny” of not only the university’s goals but the means it used to attain diversity.

Courts “must ultimately be satisfied that no workable race-neutral alternatives would produce the educational benefits of diversity,” Kennedy wrote, ordering that the appeals court consider the case more carefully in light of the stricter guidelines.

The ruling means that, at least for the moment, colleges can continue to consider race as one factor in admissions decisions, as the majority of selective public and private universities do in Massachusetts and across the country. Another case could change that approach; Justices Antonin Scalia and Clarence Thomas indicated in concurring opinions that if they were asked to do so, they would have overturned the 2003 precedent that guides colleges’ use of affirmative action.

Bill Powers, president of the University of Texas, called the decision encouraging and said it will have no immediate impact on the university’s admissions policies. He and other university presidents reiterated Monday their belief that diversity is important to the education they provide.

Harvard’s president, Drew Faust, said in a statement that she was heartened that the Supreme Court “has affirmed the vital interest of universities in bringing together students from many different backgrounds and points of view.”

Many college officials believe that there is no viable way to build a diverse class without considering race, but some said they were unsure how to definitively prove that case. Focusing on socioeconomic diversity instead of race has helped some schools, including the University of California system, maintain a diverse class after voters banned affirmative action. But maintaining that diversity has been far more difficult and expensive.

Colleges “are essentially being asked to prove a negative,” said Andrew Flagel, senior vice president for students and enrollment at Brandeis. “That does present an interesting challenge: how to prove there isn’t something else out there that would be more effective.”

Flagel said he expected the courts would come up with a rational way for colleges to make their case — and that Brandeis would not have to make major changes in its process. But “I suspect we will be parsing [the decision] for some time,” he said.

Because Kennedy set a higher bar for colleges and also faulted the lower court for deciding the case on summary judgment — without a full trial — colleges are likely to face more lawsuits and more complex, lengthy litigation, according to Robert Toone, an attorney at Foley Hoag who is preparing a seminar for the Association of Independent Colleges and Universities in Massachusetts to help officials understand what the ruling means for them.

“They spent so much time saying . . . we’re not going to defer to university administrators anymore,” said Toone, who helped write an amicus brief last summer supporting the University of Texas on behalf of the Asian American Legal Defense and Education Fund. “The message to colleges and universities is that you need to take these issues seriously, you need to get your policies in order and make sure that you are meeting constitutional standards.”

But supporters of affirmative action say the University of Texas took those concerns about as seriously as possible, compiling a report when it resumed race-conscious admissions in 2004 that documented, for example, the scarcity of minority students in small classes. In the lone dissent, Justice Ruth Bader Ginsburg argued that the university had done enough to follow the law and that the Texas policy should be upheld. (Justice Elena Kagan, who was involved in the case as solicitor general, recused herself.)

“It’s hard to know how anybody could have gone about it more carefully,” said Renee M. Landers, a professor at Suffolk University Law School, who specializes in health and constitutional law and gave a talk last fall on the potential impact on diversity among health care workers if affirmative action were curtailed in medical and nursing schools. “There’s enough in this case that any responsible university would . . . make sure it could articulate that whatever it was doing was narrowly tailored to achieve its goal.”

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